Question 1

The mean of five numbers is 20 and the sum of squared deviations from the mean is 500. What is the standard deviation?

Accepted Answer

Step 1: Variance = Sum of squared deviations ÷ Number of observations = 500 ÷ 5 = 100. Step 2: Standard Deviation = √Variance = √100 = 10. Therefore, the standard deviation is 10.

Question 2

The variance of a dataset is 64. If each value is decreased by 10, what is the new variance?

Accepted Answer

Step 1: Recall that adding or subtracting a constant from all values does not change the variance. Step 2: The variance remains 64 because variance measures spread, not location. Step 3: Therefore, the new variance is 64.

Question 3

For a dataset, if each value is multiplied by 3, how does the standard deviation change?

Accepted Answer

Step 1: Recall the property that when all values are multiplied by a constant k, the standard deviation is also multiplied by k. Step 2: If original SD = σ, new SD = 3σ. Step 3: Therefore, the standard deviation is multiplied by 3.

Question 4

The variance of a dataset is 36. What is the standard deviation?

Accepted Answer

Step 1: Recall that Standard Deviation = √Variance. Step 2: SD = √36 = 6. Therefore, the standard deviation is 6.

Question 5

A dataset has values: 2, 4, 6, 8, 10. What is the variance? (Mean = 6)

Accepted Answer

Step 1: Calculate deviations from mean (6): (2−6)² = 16, (4−6)² = 4, (6−6)² = 0, (8−6)² = 4, (10−6)² = 16. Step 2: Sum of squared deviations = 16 + 4 + 0 + 4 + 16 = 40. Step 3: Variance = 40 ÷ 5 = 8. Therefore, variance is 8.

Question 6

If the standard deviation of a set of numbers is 5, what is the variance?

Accepted Answer

Step 1: Recall that Variance = (Standard Deviation)². Step 2: Variance = 5² = 25. Therefore, the variance is 25.

Question 7

The standard deviation of a dataset is 8. If 10 is subtracted from each observation, what is the new standard deviation?

Accepted Answer

Step 1: Recall that adding or subtracting a constant from each observation does not change the standard deviation (only the mean shifts). Step 2: New standard deviation = 8 (unchanged).

Question 8

The coefficient of variation of a dataset is 25%. If the mean is 80, what is the standard deviation?

Accepted Answer

Step 1: Coefficient of Variation (CV) = (Standard Deviation / Mean) × 100%. Step 2: 25 = (SD / 80) × 100. Step 3: SD / 80 = 0.25. Step 4: SD = 80 × 0.25 = 20.

Question 9

The variance of a dataset is 36. If each observation in the dataset is multiplied by 5, what will be the new variance?

Accepted Answer

Step 1: Recall that when each observation is multiplied by a constant k, the variance is multiplied by k². Step 2: New variance = 36 × 5² = 36 × 25 = 900.

Question 10

The standard deviation of five numbers is 4. If the sum of the numbers is 50, what is the sum of their squares?

Accepted Answer

Step 1: Standard deviation σ = 4, so variance σ² = 16. Step 2: Mean = 50/5 = 10. Step 3: Use the formula Variance = (Σx²/n) - (Mean)². Step 4: 16 = (Σx²/5) - 100. Step 5: Σx²/5 = 116, so Σx² = 580.

Question 11

Two datasets A and B have the same mean of 20. Dataset A has 8 observations with variance 9, and Dataset B has 12 observations with variance 4. What is the combined variance of both datasets?

Accepted Answer

Step 1: For combined variance when means are equal, use: Var(combined) = [n₁Var₁ + n₂Var₂] / (n₁ + n₂). Step 2: Var(combined) = [8(9) + 12(4)] / (8 + 12) = [72 + 48] / 20 = 120 / 20 = 6.

Question 12

A dataset has 6 observations: 4, 6, 8, 10, 12, 14. Calculate the variance of this dataset.

Accepted Answer

Step 1: Mean = (4 + 6 + 8 + 10 + 12 + 14) / 6 = 54 / 6 = 9. Step 2: Deviations from mean: -5, -3, -1, 1, 3, 5. Step 3: Squared deviations: 25, 9, 1, 1, 9, 25. Step 4: Sum of squared deviations = 70. Step 5: Variance = 70 / 6 ≈ 11.67.

Question 13

In a dataset of 20 observations, the sum of squared deviations from the mean is 1200. A new observation with value equal to the mean is added. What is the new variance?

Accepted Answer

Step 1: Original variance = 1200/20 = 60. Step 2: When a new observation equal to the mean is added, it contributes 0 to the sum of squared deviations. Step 3: New sum of squared deviations = 1200 + 0 = 1200. Step 4: New sample size = 21. Step 5: New variance = 1200/21 ≈ 57.14.

Question 14

A dataset contains 5 observations: 12, 18, 24, 30, 36. If each observation is multiplied by 3, what is the ratio of the new standard deviation to the original standard deviation?

Accepted Answer

Step 1: When each observation is multiplied by a constant k, the standard deviation is also multiplied by k. Step 2: Original SD is multiplied by 3, so new SD = 3 × original SD. Step 3: Ratio = (3 × original SD) / original SD = 3.

Question 15

A sample of 10 students has a mean score of 60 and variance of 36. If one student with score 48 is removed, what is the new variance? (Assume the removed student's score is exactly 1 SD below the mean.)

Accepted Answer

Step 1: Original: n=10, mean=60, variance=36, so SD=6. Score 48 = 60-12 is actually 2 SDs below mean, not 1 SD. Step 2: Sum of all scores = 10×60 = 600. Step 3: Sum without the student = 600-48 = 552. Step 4: New mean = 552/9 = 61.33. Step 5: Original sum of squared deviations = 10×36 = 360. Step 6: Contribution of removed score = (48-60)² = 144. Step 7: New sum of squared deviations = 360-144 = 216. Step 8: New variance = 216/9 = 24.

Question 16

A frequency distribution has mean 40 and standard deviation 8. If all values are shifted by adding 15 to each observation, and then multiplied by 2, what is the new standard deviation?

Accepted Answer

Step 1: Adding a constant to all observations does NOT change the standard deviation (only shifts the mean). Step 2: After adding 15: new SD = 8 (unchanged). Step 3: Multiplying by 2 multiplies the SD by 2. Step 4: Final SD = 8 × 2 = 16.

Question 17

Two datasets A and B have the same mean of 50. Dataset A has variance 64 and Dataset B has variance 100. If we combine both datasets (equal number of observations from each), what is the variance of the combined dataset?

Accepted Answer

Step 1: When two datasets with equal size and same mean are combined, the combined variance equals the average of individual variances. Step 2: Combined variance = (64 + 100) / 2 = 164 / 2 = 82.

IBPS Clerk Standard Deviation & Variance

Standard Deviation & Variance— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Standard Deviation & Variance — Practice Questions

60-Second Revision — Standard Deviation & Variance